Asymptotic optimality of isoperimetric constants (Q742109)

This paper deals with the \(L^2\)-geometrically ergodic convergence in terms of the isoperimetric method. The geometric method is well known and has been used a lot in the last two decades. As illustrated by some examples in Section 5 of the paper, the method can be asymptotically optimal, even though this does not happen often. Generally speaking, the method is meaningful for general state spaces, but is actually not very practical. The isoperimetric constants are computable only in a very limited situation. This paper studies only the time-discrete Markov processes; the method is also well developed in the context of time-continuous processes. Reviewer's remark: The comments on p. 202 (in the paragraph: ``The obtained results, in particular, Theorem 3.1, carry over without changes to time-continuous Markov processes \dots{}'') seem rather misleading. Namely, although the statement ``Here, \(k_{P_n}\) has to be replaced by \(k_{P_t}\) \dots'' is logically correct, it is not useful. In the time-continuous case, a result is useful/practical if it is stated in terms of the infinitesimal generator of the process, rather than its transition probability. In this direction, the following paper is quite impressive: [\textit{Y. H. Mao} and \textit{Y. H. Song}, Acta Math. Sin., Engl. Ser. 29, No. 10, 1949--1962 (2013; Zbl 1314.60139)].